We generalize the position- and momentum-space entanglement cuts to a family of cuts corresponding to regions in the classical phase space. We explicitly compute the entanglement spectra of free fermionic many-body wavefunctions for a family of phase space entanglement cuts that continuously interpolates between position- and momentum-space cuts. For inversion symmetric wavefunctions, the phase space entanglement spectrum possess a chiral symmetry, to which a topological index can be associated.